Diagram Voronoi

In the simplest case, shown in the first picture, we are given a finite set of points {p1, …, pn} in the Euclidean plane.
In this case each site pk is simply a point, and its corresponding Voronoi cell Rk consists of every point in the Euclidean plane whose distance to pk is
less than or equal to its distance to any other pk. Each such cell is obtained from the intersection of half-spaces, and hence it is a convex polygon.
The line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices (nodes) are
the points equidistant to three (or more) sites.

The Voronoi diagram is simply the tuple of cells ( R k ) k ∈ K {\displaystyle \scriptstyle (R_{k})_{k\in K}} . In principle some of the sites can intersect and even coincide (an application is described below for sites representing shops), but usually they are assumed to be disjoint. In addition, infinitely many sites are allowed in the definition (this setting has applications in geometry of numbers and crystallography), but again, in many cases only finitely many sites are considered.

In the particular case where the space is a finite-dimensional Euclidean space, each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are convex polytopes and they can be represented in a combinatorial way using their vertices, sides, 2-dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. However, in general the Voronoi cells may not be convex or even connected.

In the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon R k {\displaystyle \scriptstyle R_{k}} is associated with a generator point P k {\displaystyle \scriptstyle P_{k}} . Let X {\displaystyle \scriptstyle X} be the set of all points in the Euclidean space. Let P 1 {\displaystyle \scriptstyle P_{1}} be a point that generates its Voronoi region R 1 {\displaystyle \scriptstyle R_{1}} , P 2 {\displaystyle \scriptstyle P_{2}} that generates R 2 {\displaystyle \scriptstyle R_{2}} , and P 3 {\displaystyle \scriptstyle P_{3}} that generates R 3 {\displaystyle \scriptstyle R_{3}} , and so on. Then, as expressed by Tran et al[3] “all locations in the Voronoi polygon are closer to the generator point of that polygon than any other generator point in the Voronoi diagram in Euclidian plane”.